The multiplication rule finds the probability that two events both occur. It multiplies the probability of the first event by the conditional probability of the second event given that the first has happened.
For an “and” problem, move through the events in sequence. Take the chance of the first step, then update for the second step based on what is already known.
Showing a random 20 of 50 problems.
Example 1
medium
Five cards are dealt. Find P(first is ace, second is king).
Example 2
easy
For independent A,B with P(A)=0.3, P(B)=0.6, find P(A∩B).
Example 3
hard
From 5 couples (10 people), pick 2 people at random. Find P(they are a couple).
Example 4
easy
Find P(A∩B) if P(A)=0.5 and P(B∣A)=0.4.
Example 5
medium
A test has P(disease)=0.01 and P(positive∣disease)=0.99. Find P(disease and positive).
Example 6
easy
P(A)=0.4 and P(B)=0.5 are independent. Find P(A∩B).
Example 7
challenge
From 5 red and 5 blue balls, draw 2 without replacement. Find P(at least one red).
Example 8
medium
A student passes math with probability 0.7. If they pass math, they pass physics with probability 0.8; otherwise with probability 0.4. Find P(passes both).Student: P(pass math)=0.7; P(pass physics | pass math)=0.8
Example 9
challenge
In a class of 23 people, find P(at least 2 share a birthday), assuming 365 equally likely birthdays.
Example 10
medium
A bag has 2 red, 3 blue, 5 green (10 total). Draw 2 without replacement. Find P(red then green).Bag: 2 red, 3 blue, 5 green — draw 2 without replacement
Example 11
easy
P(A)=0.5, P(B∣A)=0.6. Find P(A∩B).
Example 12
medium
From a deck, 3 cards are dealt without replacement. Find P(all 3 are spades).
Example 13
easy
A coin and a die: find P(tails and prime number) assuming independence (primes among 1–6 are 2,3,5).
Example 14
easy
P(A)=0.3 and P(B∣A)=0.7. Find P(A∩B).
Example 15
medium
A drawer has 6 black and 4 white socks. Draw 2 without replacement. Find P(both same color).
Example 16
medium
A factory line: part passes inspection 1 with probability 0.9, and if it passes 1 it passes inspection 2 with probability 0.8. Find P(passes both).
Example 17
easy
A bag has 4 red and 6 blue. Draw 2 without replacement. Find P(both red).
Example 18
medium
P(A∩B)=0.12 and P(A)=0.4. Find P(B∣A).
Example 19
easy
From a deck, draw 2 cards without replacement. Find P(both kings).
Example 20
easy
A box has 5 good and 1 defective fuse. Draw 2 without replacement. Find P(both good).Box: 5 good, 1 defective — draw 2 without replacement